Economic Theory (2005)
DOI 10.1007/s00199-005-0058-2
R E S E A R C H A RT I C L E
Ettore Damiano · Hao Li
Price discrimination and efficient matching
Received: June 4 2005 / Accepted: October 24 2005
© Springer-Verlag 2005
Abstract This paper considers the problem of a monopoly matchmaker that uses
a schedule of entrance fees to sort different types of agents on the two sides of
a matching market into exclusive meeting places, where agents randomly form
pairwise matches. We make the standard assumption that the match value function
exhibits complementarities, so that matching types at equal percentiles maximizes
total match value and is efficient. We provide necessary conditions and sufficient
conditions for the revenue-maximizing sorting to be efficient. These conditions
require the match value function, modified to incorporate the incentive cost of
eliciting private type information, to exhibit complementarities in types.
Keywords Complementarity · Subscription fees · Sorting structure · Random
pairwise matching · Virtual match value
JEL Classification Numbers C7 · D4
1 Introduction
Many users of Internet dating agencies such as Match.com complain about the
problem of misrepresentations and exaggerations by some users in the informaWe thank Jonathan Levin, Tracy Lewis, and the seminar audience at University of California at
Los Angeles, University of British Columbia, Duke University, and 2003 Econometric Society
North American Summer Meeting for comments and suggestions. We are also grateful for helpful
suggestions from the referees and the Editor.
E. Damiano (B) · H. Li
Department of Economics, University of Toronto, 150 St. George Street,
M5S 3G7 Toronto, Ontario, Canada
E-mail: [email protected]; [email protected]
E. Damiano and H. Li
tion they provide to the agencies.1 This problem, and the perception of it among
the public, is responsible for reducing the quality of Internet search and matching and for preventing many lonely people from fully utilizing the online dating
services, in spite of the advantages in cost, safety, anonymity and breadth of the
reach offered by the new technology compared to more traditional means of finding
dates. Although Internet dating agencies rely on individual users to report information about themselves truthfully and have little capability or resource of directly
validating the information, economic theory suggests price discrimination as a way
of making the reported information credible and improving match quality.
In this paper we look at the theoretical problem of a monopoly matchmaker that
uses a schedule of entrance fees to sort different types of agents on the two sides of
a matching market into different “meeting places,” in which agents are randomly
pairwise matched. This problem is presented in Section 2. The monopoly matchmaker faces two constraints in revenue maximization. First, the matchmaker does
not observe the one-dimensional characteristic (“type”) of each agent. This information constraint means that the matchmaker must provide incentives in terms of
match quality and fees for agents to self-select into the meeting places. We refer to
the menu of meeting places created by the matchmaker as the “sorting structure.”
Second, the monopoly matchmaker faces a technology constraint that restricts
match formation in each meeting place to random pairwise matching. This primitive matching technology allows us to focus on the impact of revenue-maximization
on the sorting structure and matching efficiency. We make the standard assumption
that the match value function exhibits complementarities between types. Under
this assumption, the “perfect sorting,” or matching types at equal percentiles with a
continuum of meeting places, maximizes the total match value and is efficient. The
goal of this paper is to understand when the perfect sorting is revenue-maximizing.
Our framework fits various two-sided market environments characterized by
sorting or self-selection based on prices.2 For example, online job search has become a major way to explore potential employer–employee relationships. However,
existing job search services such as Monster.com are plagued by job intermediaries (head hunters) that post fake entries only to collect information from job applicants and positions and then profit from the information. The job market and dating
market share a few common features that allow our framework to apply: match characteristics of market participants can be summarized in a one dimensional type; participants on one side of the market share the same preference ordering over matches
with agents on the opposite side; and types are complementary in the match value
function. Other two-sided matching markets where price-based intermediation can
potentially play an important role include matching tenants to apartments, and
matching loan applicants to bank loans. The results in the present paper show that a
monopoly matchmaker can have the same incentive as a social planner to implement
the efficient matching. In this case, the matchmaker makes directed search possible
by creating one meeting place for each type and achieves the first best matching
outcome, in spite of the technological constraint of random pairwise matching.
1
See for example The New York Times, January 18, 2001, “On the Internet, Love Really is
Blind.”
2
A limitation of this paper is the assumption of a monopoly matchmaker, as competition exists
in most two-sided markets. See our companion paper Damiano et al. (2004) for an application
of the present framework to issues of price competition.
Price discrimination and efficient matching
In Section 3 we show how the matchmaker’s problem of designing fee schedules and the corresponding sorting structure can be transformed into a problem of
monopoly price discrimination. The assumption of complementarity in the match
value function implies that the standard single-crossing condition in the price discrimination literature is satisfied for both sides of the market, and results in the
incentive compatibility constraint that a higher type receives a higher match quality. The transformation is then achieved by combining this incentive compatibility
constraint with the feasibility constraint that match qualities are generated in a
two-sided matching environment where agents participate in at most one meeting
place and are pairwise randomly matched in each meeting place. The outcome
of the transformation is a sorting structure in which the efficient matching path
in the type space (pairwise matching of types at equal percentiles) is partitioned
into pooling intervals and sorting intervals: each pooling interval on the efficient
matching path represents a meeting place with the corresponding intervals of types
on the two sides being pooled together, while each sorting interval represents a
continuum of meeting places with the types efficiently matched. This allows us to
rewrite the objective function of the monopolist by using a “virtual match value
function,” which is the match value function adjusted for the incentive costs of
eliciting private type information.
Unlike a standard price discrimination problem, the solution to the sorting structure design problem cannot be characterized by pointwise maximization, because
match qualities are not produced according to some exogenous cost function but
are instead constrained by the choice of sorting structure. In Section 4 we provide a
necessary condition for the optimal sorting structure to be the perfect sorting. This
condition requires that the virtual match value function have positive cross partial
derivatives at equal percentiles. If at any percentile it is not satisfied, the monopoly
matchmaker can increase revenue by pooling a small interval of types into a single
meeting place. This condition is local in nature, and is not generally sufficient for
the optimal sorting structure to be the perfect sorting, because a greater revenue
may be generated by pooling a large set of types on the two sides. A sufficient
condition for the perfect sorting to be optimal is that the virtual match value function is supermodular, i.e., has positive cross partial derivatives on the entire type
space. Intuitively, the inability to observe the type of agents creates an incentive
cost for the matchmaker to extract surplus because the matchmaker has to rely on
self-selection by the agents. The perfect sorting structure maximizes revenue for
the monopolist matchmaker if this incentive cost does not dominate the complementarities in the match value function. In this case, the only possible source of
inefficiency is the exclusion policy that the monopoly matchmaker may want to
adopt. For types that are matched by the monopolist, there is no distortion in match
quality provision, in contrast to the standard result in the price discrimination literature that quality is under-provided for all types except the highest. Moreover,
if the virtual match value function is supermodular, the matchmaker’s revenue is
increasing in the number of meeting places created. Hence, revenue-maximization
always leads to improvement in matching efficiency even with technological limits
on creation of meeting places.
A classical reference in the price discrimination literature is Maskin and Riley
(1984) (see also Mussa and Rosen 1978). In both the standard price discrimination
and our sorting structure design problems, the monopolist faces consumers with
E. Damiano and H. Li
one-dimensional private information about their willingness to pay, and must provide incentives for self-selection. In a price discrimination problem, the monopolist
controls the quality (or quantity) of the good provided. Consumers of different types
self-select by choosing a price-quality combination from the schedule offered by
the monopolist. In contrast, in our sorting structure design problem the monopolist
chooses a partition of the market into meeting places in which agents randomly
match, and the associated fee schedules. Besides the standard incentive compatibility and participation constraints, the monopolist also faces additional feasibility
constraints because the pair of quality schedules must be consistent with the sorting
structure.
The most closely related paper in the price discrimination literature is Rayo
(2005, forthcoming). He considers the price-discrimination problem of a monopolist that sells a status good. In his benchmark model, there is no intrinsic quality
dimension to different varieties of the good, and buyers of one variety care only
about who else are buying the same variety. Our result in Section 3 implies that this
is essentially the same price discrimination problem considered here if one restricts
to a symmetric matching environment. His results on when providing different varieties to different types is optimal can therefore be obtained as a special case of our
conditions for the perfect sorting to be optimal.
Inderst (2001) questions the classical result in the price discrimination literature that it is optimal for the monopolist to offer low types distorted contracts in
order to extract more rents from higher types. His paper looks at contract design in
a matching market environment with frictions and shows that the distortion result
does not hold anymore. In particular, for low enough search frictions all contracts
are free of distortion. The driving force of the result is that in a search and matching
environment reservation values are type dependent as higher types will generally
have more match opportunity and therefore higher reservation values. In contrast,
our no-distortion result does not rely on type-dependent reservation values, and
is generated by feasibility restrictions on match quality provision in a two-sided
matching market.
Our paper is the first to investigate intermediation in two-sided markets with
heterogeneous types and search frictions from the mechanism design point of view.
In the existing literature on two-sided search, sorting of heterogeneous types occurs
in equilibrium either because finding a good match takes time (Burdett and Coles
1997; Smith 2002), or because meeting a potential partner is costly (Morgan 1995).
Unlike these models, our framework is static and we obtain sorting as a result of
maximizing revenue by an intermediary. Building on the two-sided search literature, Bloch and Ryder (2000) analyze the problem of a monopolistic matchmaker
that competes with a decentralized matching market with frictions. Unlike our
paper, the matchmaker observes the types and can implement perfect sorting in
exchange for a fee. Due to its information advantage, the only decision for the
matchmaker is what types to service given that their reservation utilities are endogenously determined in the decentralized market.
The present paper grew out of our previous work on dynamic sorting (Damiano
et al. 2005). The two papers share the same interest in efficiency of matching markets in the presence of search frictions. In both papers, search frictions are modeled
by the primitive search technology of random meeting. In Damiano et al. (2005)
dynamic sorting provides higher types more search opportunities and improves
Price discrimination and efficient matching
matching efficiency. In the present paper, price discrimination by the monopolist
creates directed search markets and can achieve the efficient matching. In a companion paper (Damiano et al. 2004), we use a simplified framework of the present
paper to study how competition among matchmakers can affect the sorting structure
and matching efficiency.
2 The model
Consider a two-sided matching market. Without loss of generality, we assume that
the two sides have the same size. Agents of the two sides, called men and women,
have heterogeneous one-dimensional characteristics, called types. The type distribution is F (·) for men and G(·) for women. Both type distributions are assumed
to have differentiable densities, denoted as f and g, respectively. The support is
[am , bm ] for men and [aw , bw ] for women, with both subsets of IR+ , and bm and
bw possibly infinite. A match between a type x man and a type y woman produces
value xy to both the man and the woman, so 2xy is the total match value for the
pair.3 We assume that all market participants are risk neutral and have quasi-linear
preferences. They care only about the difference between the expected match value
and the entrance fee they pay. An unmatched agent gets a payoff of 0, regardless of
type. Section 5 discusses how our results can be extended when reservation utilities
either differ for the two sides or are type-dependent.
An important assumption about the matching preferences that we have made
above is that matching characteristics of each agent can be summarized in onedimensional type. This simplification relative to the reality of matching markets, facilitates comparison with the existing literature, where the assumption of
one-dimensional type is standard. Implicit in our specification of the matching
preferences is that all agents on each side of the market have homogeneous preferences. For the same price, they all prefer the highest type agents on the other side.
Clearly there are matching characteristics that are ranked differently by agents
in real matching markets. For example, it is sometimes argued that not everyone
wishes to date the smartest person. Rather, matching preferences may be single
peaked. However, when the most desirable match differs across agents, the competition among agents is reduced and so are the incentives to misrepresent this kind
of matching characteristics. Since the present paper is about how the monopoly
matchmaker uses price discrimination to mitigate the problem of misrepresentation in a matching market, we focus on matching characteristics that all agents rank
identically and compete for.4
Another important assumption about the matching preferences we have made
is that types are complementary in generating match values. This standard assumption is embedded in the match value function xy: each agent’s willingness to pay
3
Given our later assumption of 0 payoff for unmatched agents, the payoffs are unchanged if
matched couples bargain over the division of the total match value 2xy using the Nash bargaining
solution.
4
Users of online dating tend to segregate into services that cater groups that share the same preferences for non-competing characteristics. One such example is religious affiliation. Jdate.com
attracts only Jewish users while Eharmony.com targets the Christian population.
E. Damiano and H. Li
for an improvement in match type increases with the type of the agent.5 Under
this assumption, matching types at equal percentiles maximizes the total value
of pairwise two-sided matches and is efficient (Becker 1981). Formally, for each
x ∈ [am , bm ], let
sm (x) = G−1 (F (x))
be the female type at the same percentile of the male type x. We refer to
{(x, sm (x))|x ∈ [am , bm ]}
as the “efficient matching path.” We adopt the specific match value function xy for
analytical convenience. Since we allow the type distributions to be different for the
two sides of the market, this specification is without loss of generality in so far as
the match value function is multiplicatively separable and monotone in male and
female types. To be precise, any match value function of the form u(x)v(y), with u
and v being positive-valued and monotone, can be transformed into the match value
function xy by redefining types and changing the distribution functions appropriately. The separability assumption implies that each agent in a meeting place with
pairwise random matching cares only about the average agent type on the other
side, as opposed to the entire distribution. As a result, the monopolist problem of
designing the sorting structure can be reduced to be a one-dimensional problem of
match quality provision. The importance of this assumption will become clear in
Section 3. We will briefly discuss the case of non-separable match value functions
in Section 5.
A monopoly matchmaker, unable to observe types of men and women, can
create a menu of meeting places with a pair of schedules of entrance fees pm and
pw . Each man or woman participates in only one meeting place. We will restrict
each meeting place to have equal measure of men and women. We assume that
men and women in each meeting place form pairwise matches randomly, with the
probability of finding a match equal to 1 for all agents, and that the probability a
type x man meets a type y woman is given by the density of type y in that meeting
place. In other words, the meeting technology in our model is random matching.
For simplicity, we assume that meeting places cost nothing to organize. The objective function of the matchmaker is to maximize the sum of entrance fees collected
from men and women.
The technology side of our framework is modeled on the motivating example
of online dating. Imagine that each meeting place consists of two data bases, of
men and women who have paid the corresponding subscription fees. Any man
in the meeting place has access to the data base of women and can “search” it
for a match. We have assumed that the probability of finding a match is 1 for all
agents. This assumption rules out any size effect, which postulates a different probability of finding a match depending on the size of the market, and allows us to
focus on the issue of price discrimination. The search technology in each meeting
place, which is pairwise random matching, is admittedly primitive, compared to
5
In online dating, a more attractive individual is more likely to have a successful first date
than a less attractive individual, so even if both derive the same utility from a given potential
match, the more attractive individual is willing to pay more for an improvement in the quality of
the potential match.
Price discrimination and efficient matching
the actual matching technology used by online dating services where agents can
search according to the information available on the data base and exchange further information through anonymous email correspondence. We have adopted the
pairwise random matching technology in order to focus on the misrepresentation
problem, by implicitly assuming that any information volunteered by participants
beyond what is signaled by their choices of meeting place is not credible and
therefore cannot be used to improve matching efficiency. The importance and the
implications of the assumption of random pairwise matching are discussed in Section 5. Similarly, we have ignored the possibility of verifying certain information
by providers of online dating services. For example, claims of college education
in principle can be verified. Verifiable information can help the monopolist extract
surplus. In the extreme case where the type information is public, the monopolist
can achieve perfect discrimination through the perfect sorting. In general, the way
availability of verifiable information changes the conditions for the optimality of
the perfect sorting depends on how conditioning on public information affects the
type distributions. We focus on unverifiable information and misrepresentation.
We refer to a menu of meeting places as a sorting structure. Let φm be a set-valued function that maps any male type x in [am , bm ] to a subset φm (x) of [aw , bw ].
The set φm (x) represents the set of female types that the male type x men can
hope to meet. We sometimes refer to φm (x) as type x men’s “match set.” We allow
the possibility that male type x is excluded by the monopolist matchmaker, with
φ(x) = ∅. Define φw similarly, and denote φ = φm , φw . For any X ⊆ [am , bm ],
let
m (X) = {y|y ∈ φm (x) for some x ∈ X}
represent the union of match sets of male types in X. Define w similarly.
Definition 2.1 A sorting structure φ is feasible if for any x, x̃ ∈ [am , bm ], y, ỹ ∈
[aw , bw ], X ⊆ [am , bm ] and Y ⊆ [aw , bw ], i) y ∈ φm (x) implies x ∈ φw (y), and
x ∈ φw (y) implies y ∈ φm (x); ii) φm (x) = φm (x̃) implies φm (x) ∩ φm (x̃) = ∅, and
φw (y) = φw (ỹ) implies φw (y) ∩ φw (ỹ) = ∅; and iii) m (X) has the same measure
as {x|φm (x) ⊆ m (X)}, and w (Y ) has the same measure as {y|φw (y) ⊆ w (Y )}.
Condition (i) is analogous to the standard symmetry condition for matching
correspondences. It states that if type x men are participating in a meeting place
where there are type y women, then type y women are participating in a meeting
place where there are type x men, and vice versa. This condition is needed for
a meeting place to have the interpretation of a matching market. Condition (ii)
requires that each type participates in at most one meeting place. This simplifies
the analysis. Condition (iii) requires that each meeting place consists of men and
women of equal measures. This ensures that match probability is one for each agent
in any meeting place, and helps us minimize the role of search technologies and
focus on the impact of revenue-maximization on the sorting structure and matching
efficiency.
3 Weak sorting
The monopolist’s problem is to choose a sorting structure and the corresponding
two fee schedules, one for males and one for females. A sorting structure assigns
E. Damiano and H. Li
to each male type a set of potential female matches and to female types a set of
potential male partners. The design problem appears multi-dimensional because
what a type buys from the matchmaker is a type distribution on the other side of
the meeting place. However, the assumption of a multiplicatively separable match
value function allows us to reduce the problem to one dimension. Our first step of
analysis is to substitute a pair of expected match types for each meeting place in
the design problem, and transform the market design problem to a more familiar
price discrimination problem.
A feasible sorting structure φ generates two schedules of expected match types,
qm and qw . The function qm : [am , bm ] → [aw , bw ] ∪ {0} assigns to each male type
the expected value of his match; the function qw : [aw , bw ] → [am , bm ] ∪ {0} is
the corresponding function for female types. We refer to q = qm , qw as a pair of
“quality schedules,” given by6
qm (x) = E[y|y ∈ φm (x)]; qw (y) = E[x|x ∈ φw (y)]
(1)
for all x ∈ [am , bm ] such that φm (x) = ∅ and y ∈ [aw , bw ] such that φw (y) = ∅.
We adopt the convention that if any type is excluded by the matchmaker, the match
quality assignment is 0, which is the reservation utility. The lemmas in the remainder of this section refer to types that are served by the monopoly matchmaker. With
the convention we have adopted, the lemmas can be easily restated to cover the
excluded types.
As in a price discrimination problem, the monopolist does not observe agent
types and must rely on self-selection of agents into their assigned expected match
quality.7 Given equations (1), we can now formally state the optimal mechanism
design problem of the matchmaker. Let pm (x) be the participation fee for type
x and define pw (y) similarly; denote p = pm , pw . The monopolist chooses a
feasible sorting structure φ and a pair of fee schedules p to maximize the revenue
bm
am
pm (x) dF (x) +
bw
pw (y) dG(y),
aw
subject to incentive compatibility constraints
xqm (x) − pm (x) ≥ xqm (x̃) − pm (x̃); yqw (y) − pw (y) ≥ yqw (ỹ) − pw (ỹ)
for all x, x̃ ∈ [am , bm ] and y, ỹ ∈ [aw , bw ] respectively, and participation constraints
xqm (x) − pm (x) ≥ 0; yqw (y) − pw (y) ≥ 0
for all x ∈ [am , bm ] and y ∈ [aw , bw ] respectively, where q is given in (1).
6
Without the restrictions of types participating in at most one meeting place, φ would not be
sufficient to define qm and qw and we would need additional notation to specify the fraction of
agents of a given type who participate in any given meeting place.
7
Since the matching market is two-sided, self-selection involves a coordination problem that
is absent in a standard price discrimination problem. We ignore such problem in this paper by
assuming that the monopoly matchmaker can decide how agents self-select so long as the sorting
structure is feasible and incentive compatible. See our companion paper (Damiano et al. 2004)
for a discussion of how to resolve the coordination problem.
Price discrimination and efficient matching
Under the complementarity assumed in the match value function, standard
arguments imply that qm being nondecreasing is necessary for the incentive compatibility constraints for men to be satisfied (see, e.g., Maskin and Riley 1984).
Further, the associated indirect utility Um (x) of male type x, defined as
Um (x) = xqm (x) − pm (x),
satisfies the envelope condition
Um (x) = qm (x).
(2)
at every x such that qm (x) is continuous. Finally, condition (2) and the monotonicity of qm together are sufficient for incentive compatibility. Similar observations
hold for monotonicity of qw and the indirect utility function Uw of women.
Unlike in a typical price discrimination problem, the monopolist can only
choose schedules qm and qw consistent with some feasible sorting structure. Through
a series of lemmas, whose proofs can be found in the Appendix, we show how the
feasibility constraints on the sorting structure translate into direct restrictions on
quality schedules. Monotonicity of any schedules q leads to the following definition.8
Definition 3.1 An interval Tm ⊆ [am , bm ] is a maximal pooling interval under qm
if qm is constant on Tm , and there is no interval Tm ⊃ Tm such that qm is constant
on Tm .
Maximal pooling intervals Tw under qw can be similarly defined. We say that
q = qm , qw is “feasible” if there is a feasible φ = φm , φw such that equations
(1) are satisfied for almost all x and y. We call φ the “associated” sorting structure.
Lemma 3.2 If q is feasible, then for any maximal pooling interval Tm under qm
and any associated sorting structure φ, m (Tm ) is a maximal pooling interval
under qw .
By symmetry, if a pair of nondecreasing schedules q is feasible, then w (Tw )
is a maximal pooling interval under qm for any maximal pooling interval Tw under qw and any associated sorting structure φ. A corollary of Lemma 3.2 is thus
w (m (Tm )) = Tm , and symmetrically m (w (Tw )) = Tw . Another implication
is that for any associated sorting structure φ, and for any maximal pooling interval
Tm under qm , we have qm (x) = E[y|y ∈ m (x)] for all x ∈ Tm .9 Symmetrically, for any maximal pooling interval Tw under qw and for any y ∈ Tw , we have
qw (y) = E[x|x ∈ w (Tw )].
Lemma 3.2 is the first step in showing that a pair of nondecreasing, feasible
schedules q defines two sequences {Tml }Ll=1 and {Twl }Ll=1 of maximal pooling intervals in [am , bm ] and [aw , bw ] respectively, with Twl = m (Tml ) and Tml = w (Twl )
8
There is no need to specify whether a maximal pooling interval contains the two end points.
The assignment of values of qm and qw to the end points does not affect the revenue function
stated later in Proposition 3.6.
9
In general, it is not true that φm (x) = m (Tm ) for all x ∈ Tm , as there can be more than one
way of assigning match sets for x in Tm such that qm (x) is constant. However, by condition (iii) of
Definition 2.1, we have Tm E[y|y ∈ φm (x)] dF (x)(F (sup(Tm )−F (inf(Tm )))E[y|y ∈ m (Tm )].
Since E[y|y ∈ φm (x)] equals qm (x) and is constant on Tm , it follows that qm (x) = E[y|y ∈
m (Tm )] for all x ∈ Tm .
E. Damiano and H. Li
for each l. The next step is to identify the end points of each maximal pooling
interval.
Lemma 3.3 If q is feasible, then for any maximal pooling interval Tm under qm and
any associated sorting structure φ, sm (inf Tm ) = inf m (Tm ) and sm (sup Tm ) =
sup m (Tm ).
Lemmas 3.2 and 3.3 completely characterize a nondecreasing, feasible q for x
and y in maximal pooling intervals. It remains to characterize qm (x) and qw (y) for
any x and y not in a maximal pooling interval, respectively.
Lemma 3.4 If q is feasible, then qm (x) = sm (x) for any x ∈ [am , bm ] such that x
does not belong to any maximal pooling interval under qm .
The following proposition summarizes the feasibility restrictions on incentive
compatible quality schedules that we have derived from the restrictions imposed
on feasible sorting structure (Definition 2.1). For any am ≤ x < x̃ ≤ bm , let
µm (x, x̃) = E[t|x ≤ t ≤ x̃]
be the mean male type on the interval [x, x̃]. Define µw (y, ỹ) similarly.
Proposition 3.5 A pair of nondecreasing quality schedules qm , qw is feasible if
and only if i) for any maximal pooling interval Tm under qm and each x ∈ Tm ,
qm (x) = µw (sm (inf Tm ), sm (sup Tm )) and qw (sm (x)) = µm (inf Tm , sup Tm ); and
ii) for any x not in any maximal pooling interval under qm , qm (x) = sm (x) and
qw (sm (x)) = x.
Proof Necessity of (i) and (ii) follow immediately from Lemmas 3.2–3.4. For
sufficiency, fix any q that is nondecreasing and feasible. For each maximal pooling interval Tm under qm , construct the set-valued function φm such that φm (x) =
[sm (inf Tm ), sm (sup Tm )] for any x in the closure of Tm , and correspondingly φw
such that φw (y) = [inf Tm , sup Tm ] for any y ∈ [sm (inf Tm ), sm (sup Tm )]. For all
other x, let φm (x) = {sm (x)} and φw (sm (x)) = {x}. By conditions i) and ii) stated
in the proposition, φm (x) and φw (y) are well-defined for all x ∈ [am , bm ] and
y ∈ [aw , bw ] respectively, and further, φm and φw satisfy equations (1) for almost
all x and y. Thus, qm , qw is feasible.
The above result can be viewed as a characterization of any feasible sorting
structure associated with an incentive compatible, feasible pair of quality schedules. We refer to the characterization as “weak sorting.” Since meeting places are
mutually exclusive in type, if two types on the same side of the market participate
in two different meeting places, the higher type not only has a higher average match
type, but never gets a lower match.
We have completed transforming the design problem from choosing a feasible
and incentive compatible sorting structure φm and φw to a problem of choosing a
pair of nondecreasing quality schedules that satisfy Proposition 3.5. The advantage of this transformation is that from the mechanism design literature we know
how to manipulate the incentive compatibility and individual rationality constraints
Price discrimination and efficient matching
associated with one-dimensional schedules q to rewrite the matchmaker’s revenue.
Define
Jm (x) = x −
1 − G(y)
1 − F (x)
, Jw (y) = y −
g(y)
f (x)
to be the “virtual type” for male type x and female type y, respectively. Virtual
type of x takes into account the incentive cost of eliciting private type information
from type x. These are familiar definitions from the standard price discrimination
literature (e.g., Myerson 1981). Next, we combine the virtual types and define
K(x, y) = xJw (y) + yJm (x)
(3)
as the “virtual match value” for male type x and female type y. Virtual match
value of types x and y is based on the match value between x and y with proper
adjustment of the incentive costs of eliciting truthful information from the two
types.
For the following proposition, note that for any q that is nondecreasing and feasible, there are at most countable many maximal pooling intervals. This is because
for any maximal pooling interval Tm , the quality schedule qm is discontinuous at
inf Tm (unless inf Tm = am ) and sup Tm (unless sup Tm = bm ). Since qm is monotone, it can only have a countable number of discontinuities. Let L be the total
number of maximal pooling intervals under qm ; note that we allow L to be infinite.
Proposition 3.6 Fix a pair of nondecreasing and feasible quality schedules qm ,
qw . Define cm = inf{x : qm (x) > 0}, and let {Tml }Ll=1 be the collection of all maximal pooling interval under qm over [cm , bm ]. The maximum revenue generated by
qm , qw is
K(x, sm (x)) dF (x)
[cm ,bm ]\(∪Ll=1 Tml )
+
L l=1
sup Tml
inf
Tml
sm (sup Tml )
sm (inf
Tml )
K(x, y) dG(y) dF (x)
.
F (sup Tml ) − F (inf Tml )
(4)
Proof Incentive compatibility and feasibility of the quality schedules imply that
the monopoly matchmaker’s exclusion policy takes the form of a cutoff male type
cm ∈ [am , bm ] such that male types x < cm and female types y < sm (cm ) are
excluded. Using the definition of the indirect utility function Um , we can write the
total revenue from the male side as
bm
(xqm (x) − Um (x)) dF (x).
cm
After integration by parts we can use (2) and the definition of virtual type function
Jm to rewrite the revenue from the male side as
bm
qm (x)Jm (x) dF (x).
(5)
−Um (cm )(1 − F (cm )) +
cm
E. Damiano and H. Li
The revenue from the female side can be similarly stated. The cutoff types cm and
sm (cm ) receive their reservation utility of 0 in any optimal price discrimination
mechanism for the monopolist. The revenue formula (4) in Proposition 3.6 then
follows from equation (3) and the characterization result of Proposition 3.5.
Proposition 3.6 restates the original sorting structure design problem given at
the beginning of this section as choosing quality schedules q. We note that there
are two components in the restated maximization problem: one is the exclusion
policy or choosing cm , and the other is the optimal sorting problem for a given
cm . By assumption, the match value of any pair of types is positive and the reservation utility of each type is zero, implying that a social planner that maximizes
the total match value will implement full market coverage. In contrast, the virtual
match value of a pair of types need not be positive, and so the monopolist may
find it optimal to exclude some types. The focus of the present paper is when the
revenue-maximizing sorting structure is the perfect sorting, which can be studied
independently of the exclusion policy. The conditions for the perfect sorting to
be revenue-maximizing, derived in the next section, are a characterization of the
optimal sorting problem for any given exclusion policy.
The objective function of the optimal sorting problem is given by the revenue
formula (4). The formula contains two terms, corresponding respectively to the
revenue from the types that are perfectly sorted and the revenue from a sequence
of pooling regions. Note that the quality schedule q does not appear explicitly in
the revenue formula; by Proposition 3.5, the feasibility constraint on q, together
with the incentive compatibility constraint, has already pinned down q once the
sequence of maximal pooling intervals {Tml }Ll=1 is given. Thus, the monopolist’s
optimal sorting problem is reduced to choosing the break points of the maximal
pooling intervals.10 We can think of the monopolist partitioning the set of serviced
male types [cm , bm ] into a sequence of pooling intervals and sorting intervals, with
the set of serviced female types correspondingly partitioned. Since the revenue
is written as sum of revenues from these intervals in the formula of Proposition
3.6, whether it is optimal to pool or to sort the types in one particular interval
can be determined in isolation. This feature will be repeatedly used in the next
section.
4 Perfect Sorting
Proposition 3.5 establishes weak sorting as the outcome of satisfying both the
incentive compatibility constraint in self-selection and the feasibility constraint on
the sorting structure. Weak sorting can take different forms, from pooling the entire population of agents into a single meeting place to segregating each type into
separate meeting places. Due to the assumption of complementarity in the match
10
By definition two sorting intervals cannot be adjacent to each other. However, it is possible,
and may even be optimal, to have two pooling intervals adjacent to each other.
Price discrimination and efficient matching
value function xy, the finer the agents are partitioned into separate meeting places,
the higher the matching efficiency in terms of the total match value.11 The question
is whether the monopoly’s revenue is also increased.
In this section, we use the revenue formula of Proposition 3.6 to study the
implications of revenue maximization to the sorting structure and matching efficiency. We are particularly interested in the perfect sorting structure. If the perfect
sorting maximizes the monopolist’s revenue, then the monopolist has the same
incentive to create meeting places as a social planner who maximizes the total
match value. In this case, the incentive cost of eliciting private type information
generates no distortion in terms of match quality provision. This is in contrast with
the standard price discrimination result that quality is under-provided for all types
but the very highest (Mussa and Rosen 1978; Maskin and Riley 1984). The standard result is commonly explained in terms of the tradeoff between “efficiency”
and rent extraction: efficiency for a given type means a quality level that maximizes the trade surplus, defined as the type’s utility of consuming the quality
less the cost of producing it, but the price function that implements the efficient
quality schedule leaves too much rent to types. This tradeoff is resolved by a downward distortion of the quality level for all types except the highest. Indeed, in the
standard model the efficient quality schedule is never profit-maximizing, because
marginally lowering the qualities while maintaining full separation of types will
have a first order effect on information rent and only a second order effect on efficiency, and will thus increase the revenue. This argument does not work in our
model where quality distortion can only be achieved by pooling types due to the
feasibility constraints (equations 1) on the quality schedules. The choice of the sorting structure uniquely determines the monopolist’s quality decisions, and hence
the efficiency and the rent. Whether or not the monopolist optimally chooses the
efficient quality schedules is equivalent to whether or not the perfect sorting is the
solution to the monopolist’s optimal sorting problem. First, we provide a necessary
condition.
Proposition 4.1 Suppose that a pair of quality schedules q is optimal and qm (x) =
sm (x) for all x ∈ (x1 , x2 ). Then, for all x ∈ (x1 , x2 )
K12 (x, sm (x)) = Jm (x) + Jw (sm (x)) ≥ 0.
(6)
Proof Fix some x ∈ (x1 , x2 ) and a small positive . Construct a new pair of quality
schedules q() by pooling the male types on the interval [x, x + ] with the female
types on the corresponding interval [sm (x), sm (x + )], while retaining the sorting
structure outside the region [x, x + ] × [sm (x), sm (x + )] and the quality schedules. Let x () be the revenue difference between the original quality schedules
and q(). We note that q() is nondecreasing by construction, and feasible because
it satisfies Proposition 3.5. Thus, we can apply the revenue formula of Proposition
11
Although this statement is intuitively obvious, we are not aware of a direct proof in the
existing literature, except that McAfee (2002) shows that a relatively large efficiency gain can
be made by optimally splitting one market into two. Proposition 4.4 below provides a general
argument for the efficiency gain, if we replace K with the match value function xy in the proof of
the proposition and note that by assumption the match value function satisfies supermodularity.
E. Damiano and H. Li
3.6 and write x () as:
x+
x
x+
K(t, sm (t)) dF (t) −
sm (x+)
K(t, y)
x
sm (x)
dG(y) dF (t)
.
F (x + ) − F (x)
Consider the behavior of x () for → 0. Clearly, we have x (0) = 0.
Straightforward but tedious calculations12 reveal that x (0) = x (0) = 0, while
1
x (0) = 2 (Jm (x) + Jw (sm (x)))f (x)sm (x). If (6) is not satisfied at x, then there
exists an > 0 such that the monopoly matchmaker can increase revenue by pooling male types in [x, x + ] and corresponding female types in [sm (x), sm (x + )]
into a single meeting place, instead of perfectly sorting these types.
In a simple price-quality discrimination problem, where the trade surplus equals
the product of the quality and the type less the cost of producing the quality, one
presumes full separation of types, drops the monotonicity constraint on the quality
schedule, and chooses a quality level to maximize the “virtual surplus” for each
type, which is the trade surplus with the virtual type in place of the type. This pointwise maximization method cannot be applied in our mechanism design problem.
This is because match qualities are not produced according to some exogenous
cost function: in equation (5), pointwise maximization would imply exclusion (i.e.
qm (x) = 0) for any type x with negative virtual type Jm (x), and unbounded quality
if Jm (x) is positive. Instead, quality provision for the two sides of the market is
simultaneously determined by the choice of sorting structure through Proposition
3.5. This allows us to use the local approach to identify the necessary condition
for the perfect sorting to be optimal.13 Note that when the two sides of the market
have the same type distributions, with sm (x) = x, condition (6) boils down to the
virtual type function being nondecreasing (although monotonicity of the virtual
types on both male and female sides is clearly not required with asymmetric distributions.) In the simple price discrimination problem, monotonicity of the virtual
type is necessary for a strictly increasing quality schedule to be optimal. Although
our conclusion coincides with the standard monotonicity condition in the special
case of identical type distributions for the two sides, the logic is different between
the two models. In the price discrimination problem, monotonicity of the virtual
type is equivalent to monotonicity of the solution to the pointwise maximization
problem, and is necessary for the solution to be incentive compatible. In contrast,
the necessity of the local condition (6) follows from a variational argument over
the revenue formula (equation 4), which respects the feasibility constraint as well
as the monotonicity constraint on the quality schedules.
The local necessary condition (6) in Proposition 4.1 does not impose any constraint on the behavior of the virtual match value function away from a small
neighborhood of the efficient matching path. As a result, it fails to ensure that a
greater revenue cannot be generated by pooling a large set of types. A sufficient
condition for the perfect sorting to be optimal is that the virtual match value function
12
The details are available from the authors upon request.
Bergemann and Pesendorfer (2001) use similar techniques to answer the question of how
much private information an auctioneer should allow the bidder to learn about his valuation. The
analogy between our sorting structure design problem and theirs can be seen if one thinks of a
partition element in an information structure in their paper as a pooling of types in our problem.
13
Price discrimination and efficient matching
is supermodular on the entire type space, that is,
K12 (x, y) = Jm (x) + Jw (y) ≥ 0
(7)
for all x ∈ (cm , bm ) and y ∈ (cw , bw ).14 If F and G are drawn from the large
class of distribution functions that satisfy the standard condition of non-decreasing
hazard rate, then K is supermodular.
Proposition 4.2 If K satisfies (7) for any x ∈ (x1 , x2 ) and y ∈ (sm (x1 ), sm (x2 ))
with strict inequalities, then any nondecreasing, feasible pair of quality schedules
with (x1 , x2 ) × (sm (x1 ), sm (x2 )) as the interior of a maximal pooling region is
non-optimal.
Proof Let q̂ = q̂m , q̂w be a pair of nondecreasing and feasible quality schedules, with (x1 , x2 ) × (sm (x1 ), sm (x2 )) as the interior of a maximal pooling region.
Construct a pair of quality schedule q ∗ = qm∗ , qw∗ such that: i) qm∗ and qw∗ are
identical to q̂m and q̂w outside the maximal pooling intervals that contain (x1 , x2 )
and (sm (x1 ), sm (x2 )), respectively; and ii) qm∗ (x) = sm (x) and qw∗ (sm (x)) = x for
any x in the maximal pooling intervals that contain (x1 , x2 ) and (sm (x1 ), sm (x2 )),
respectively. By construction, q ∗ is nondecreasing and feasible. Let denote the
revenue difference between q ∗ and q̂, given by
x2 sm (x2 )
x2
dG(y) dF (x)
.
(8)
K(x, sm (x)) dF (x) −
K(x, y)
=
F (x2 ) − F (x1 )
sm (x1 )
x1
x1
The first term on the right-hand-side of the above can be also written as
x2 sm (x2 )
x2
dG(y) dF (x)
.
K(x, sm (x)) dF (x) =
K(x, sm (x))
F
(x2 ) − F (x1 )
x1
x1
sm (x1 )
With a change of variable y = sm (x), we can also write
x2 sm (x2 )
x2
dG(y) dF (x)
.
K(x, sm (x)) dF (x) =
K(sm−1 (y), y)
F
(x2 ) − F (x1 )
x1
x1
sm (x1 )
In a similar way, after two changes of variable x = sm−1 (ỹ) and y = sm (x̃), the
second term on the right-hand-side of (8) can be written as
x2 sm (x2 )
x2 sm (x2 )
dG(y) dF (x)
dG(y) dF (x)
=
.
K(x, y)
K(sm−1 (y), sm (x))
F (x2 ) − F (x1 ) x1 sm (x1 )
F (x2 ) − F (x1 )
x1 sm (x1 )
Hence, is equal to one-half of
14
The proof of Proposition 4.2 below remains valid if K(x, sm (x)) + K(x̃, sm (x̃)) ≥
K(x, sm (x̃)) + K(x̃, sm (x)) for all x, x̃ ∈ (x1 , x2 ). Therefore, this weaker condition is also
sufficient for the perfect sorting to be optimal. When the virtual match value function K is twice
differentiable, supermodularity of K requires minx Jm (x) + miny Jw (y) ≥ 0, while the weaker
condition implies minx Jm (x) + Jw (sm (x)) ≥ 0. If the efficient matching path sm (x) is linear,
including the special case where F and G are identical so that sm (x) = x, then the minimum of
Jm (x) and the minimum of Jw (y) are achieved at a point (x, y) on the efficient matching path
and therefore supermodularity and the weaker condition coincide. A linear efficient matching
path occurs only when the distribution of type on one side of the market is the same as the distribution of a linear transformation of type on the other side. In general K can satisfy the weaker
condition but fail to be supermodular. Examples with simple distribution functions that illustrate
the difference between the two concepts are available upon request.
E. Damiano and H. Li
x2
sm (x2 )
sm (x1 )
x1
K(x, sm (x)) + K(sm−1 (y), y) − K(x, y) −
dG(y) dF (x)
,
K(sm−1 (y), sm (x))
F (x2 ) − F (x1 )
which is strictly positive because K satisfies (7) with strict inequalities.
The idea of the proof comes from the revenue formula in Proposition 3.6. The
revenue from perfectly sorting the types in the region (x1 , x2 ) × (sm (x1 ), sm (x2 )) is
the integral of the virtual match value function K along the segment of the efficient
matching path {(x, sm (x))|x ∈ (x1 , x2 )}, while the revenue from pooling the types
is the integral of K over the entire region. By changes of variables we can write the
revenue difference as the integral of a function which has positive values because
K is supermodular. Proposition 4.2 immediately implies a sufficient condition for
the perfect sorting to be optimal.
Corollary 4.3 If K satisfies (7) for any x ∈ (cm , bm ) and y ∈ (cw , bw ), then the
perfect sorting is optimal.
Proposition 4.2 suggests that when the virtual match value function K satisfies
(7) for all x and y in some region (x1 , x2 ) × (sm (x1 ), sm (x2 )), breaking up the
region into sufficiently many small pooling regions generates more revenue than
pooling all types in the region together. The next proposition establishes that under
supermodularity of K we in fact have a stronger result that every division of the
market into meeting places increases the revenue. This is useful in practice because
it implies that setting up a new meeting place always strictly increases revenue.
Proposition 4.4 Let q ∗ be a pair of nondecreasing, feasible quality schedules
with (x1 , x2 ) × (sm (x1 ), sm (x2 )) as the interior of a maximal pooling region. If K
is strictly supermodular on (x1 , x2 ) × (sm (x1 ), sm (x2 )), then for any t ∈ (x1 , x2 ),
any pair of nondecreasing, feasible quality schedules q̂ such that q̂ is identical
to q ∗ outside [x1 , x2 ] × [sm (x1 ), sm (x2 )] and q̂ has (x1 , t) × (sm (x1 ), sm (t)) and
(t, x2 ) × (sm (t), sm (x2 )) as the interiors of two maximal pooling regions generates
strictly more revenue than q ∗ .
Proof Let the revenue difference between q̂ and q ∗ be . Using the revenue formula from Proposition 3.6 we can show that is proportional to
t
x1
sm (t)
sm (x1 )
−
t
x2
K(x, y) dFl (x) dGl (y) +
sm (x2 )
K(x, y) dFh (x) dGh (y)
t
sm (t)
x2
sm (x2 )
sm (t)
K(x, y) dFh (x) dGl (y), (9)
K(x, y) dFl (x) dGh (y)−
x1 sm (t)
t
sm (x1 )
where for any x ∈ (x1 , t) and x̃ ∈ (t, x2 )
Fl (x) =
F (x)
,
F (t) − F (x1 )
Fh (x̃) =
F (x̃)
,
F (x2 ) − F (t)
Price discrimination and efficient matching
and for y ∈ (sm (x1 ), sm (t)) and any ỹ ∈ (sm (t), sm (x2 ))
Gl (y) =
G(y)
,
F (t) − F (x1 )
Gh (ỹ) =
G(ỹ)
.
F (x2 ) − F (t)
Next, apply the change of variables Fh (x) = Fl (x̃) to x in the second integral and
in the fourth integral, and Gh (y) = Gl (ỹ) to y in the second integral and in the
third integral. Then, is proportional to the double integral of
−1
−1
K(x, y)+K(Fh−1 (Fl (x)), G−1
h (Gl (y)))−K(x, Gh (Gl (y)))−K(Fh (Fl (x)), y),
which is strictly positive because Fh−1 (Fl (x)) > x, G−1
h (Gl (y)) > y and K is
strictly supermodular on (x1 , x2 ) × (sm (x1 ), sm (x2 )).
The revenue difference between sorting the types in (x1 , x2 ) × (sm (x1 ), sm (x2 ))
into two meeting places and pooling all types in the region, is the integral of a
function with x varying between x1 and t and correspondingly y between sm (x1 )
and sm (t). This function has positive values due to supermodularity of K. Note
that other changes of variables would also work. For example, one can define new
integration variables by setting Fh (x̃) = 1 − Fl (x) and Gh (ỹ) = 1 − Gl (y). The
proof of the proposition proceeds in a similar fashion; the only change is that the
revenue difference is the integral of a different function, which is still positively
valued due to supermodularity of K.
Equation (9) implies that a necessary condition for (x1 , x2 ) × (sm (x1 ), sm (x2 ))
to be the interior of a maximal pooling region is that there does not exist a point
(t, sm (t)) on the efficient matching path contained in the region such that the virtual
match value function is “on average” supermodular at the point. An implication
of this result is that if the matchmaker can create at least two meeting places, it
would never be optimal to pool all men and women into a single market. This follows because regardless of the type distributions, the virtual type functions Jm and
Jw eventually become increasing towards the end of the efficient matching path
and reach their respective maximum at the end. This in turn implies that there is
always a point (t, sm (t)) such that the virtual type functions satisfy minx≥t Jm (x) ≥
maxx≤t Jm (x) and miny≥sm (t) Jw (y) ≥ maxy≤sm (t) Jw (y), and therefore the virtual
match value function is supermodular on average at (t, sm (t)). At this point it would
increase the revenue to split the market into two pools.
To conclude this section, we explicitly calculate the fee schedules under the perfect sorting. Using condition (2) and the perfect sorting condition qm (x) = sm (x),
we have
x
sm (t) dt.
pm (x) = xsm (x) −
cm
A similar expression holds for the female fee schedule pw . Note that pm and pw are
continuous. This property holds only when the perfect sorting is optimal. In general
any pooling will make the quality schedule discontinuous. Since the indirect utility
functions are necessarily continuous, the fee schedules will be discontinuous at the
boundaries of each maximal pooling region. Finally, the sum of revenues from a
pair of male and female types on the efficient sorting path is
pm (x) + pw (sm (x)) = 2xsm (x) − Um (x) − Uw (sm (x)).
E. Damiano and H. Li
This implies that the rate of increase of the sum of revenues is
pm
(x) + pw (sm (x))sm (x) = sm (x) + xsm (x),
which is one-half of the rate of increase of the sum of match values 2xsm (x).
Thus, even though it is optimal for the monopoly matchmaker to implement the
socially efficient sorting structure, the monopolist cannot implement perfect price
discrimination and does not extract all the surplus.
5 Discussions
So far we have considered conditions for the perfect sorting to be optimal under two
substantive assumptions about the reservation utility. First, we have assumed that
the reservation utility is the same for the two sides of the market. This assumption
can be easily dispensed without affecting our results. Given any exclusion policy
(i.e., cutoff types cm and cw such that cw = sm (cm )), the solution to the optimal
sorting problem is independent of the reservation utilities, because the only change
to the objective function (the revenue formula 4) is the addition of two constant
terms −U m (1 − F (cm )) and −U w (1 − G(cw )), where U m and U w are the reservation utility for men and for women, respectively. Thus, the conditions for the
perfect sorting to be optimal will not change.15
The second assumption is that the reservation utility is type independent. However, higher types may have better outside options. This can be captured by assuming that men and women excluded from the monopolist’s mechanism can randomly
match among each other for free. In this case the reservation utility of a type is the
type’s expected payoff from joining the free pool, and is endogenously determined
by the exclusion policy of the monopolist. Under any feasible, incentive compatible market structure, the types that participate in the free pool are determined by
a cutoff rule, with only men and women below the respective cutoff types participating in the free pool. This is because the free pool corresponds to a participation
fee of zero, so it cannot be optimal for the monopolist to create a meeting place
with a quality lower than the quality of the free pool. Further, as in the case of
exogenous type-independent reservation utility, the fees for the types served by
the matchmaker are determined by the usual incentive compatibility constraints,
rather than by the participation constraint that these types have to get as much utility from the matchmaker as from the free pool, even though higher types receive
more utility from the free pool. This latter claim follows from the fact the indirect
utility of a type x above the cutoff increases at the rate of qm (x) (equation 2),
while the utility from the free pool increases at the rate of the conditional mean of
female types below the cutoff, which is lower than qm (x). Thus, for any exclusion
policy or a pair of cutoff types cm and cw , the introduction of the free pool (with
the utility for unmatched agents remaining zero) changes the objective function
(the revenue formula 4) by adding two constants −cm µw (aw , cw )(1 − F (cm )) and
−cw µm (am , cm )(1 − G(cw )). This means that the solution to the optimal sorting
15
Asymmetric reservation utilities will in general change the optimal exclusion policy. For
example it might be optimal to charge a negative price to the cutoff type on the side with a higher
reservation utility in order to induce greater participation and extract more rent from the other
side.
Price discrimination and efficient matching
problem does not change as a result of endogenous reservation utility, and the
conditions for the perfect sorting to be optimal remain unchanged.16
An important assumption in our model is that the match value function is multiplicatively separable. Without this assumption, the payoff to an agent from a random
pairwise matching in a meeting place generally depends on the entire type distribution of participants from the other side. This means that the monopolist problem
of designing the sorting structure φ cannot be reduced to be a one-dimensional
problem of choosing quality schedules q. In place of equations (1), the monopolist
has to choose a pair of “match schedules” αm and αw , with αm (x) representing the
distribution of female types on the match set φm (x) for male type x. The key to the
weak sorting result of Proposition 3.5 is the monotonicity condition on the quality
schedule, but there is no counterpart to this ordering with a non-separable match
value function because αm (x) is a multi-dimensional object. Thus, we cannot further reduce the monopolist problem of designing the sorting structure to choosing
the break points of maximal pooling intervals. However, if one is willing to assume
weak sorting, then we can derive an analogous expression for the revenue formula
of Proposition 3.6, and identify necessary and sufficient conditions for the perfect
sorting to be optimal in the same way as in Section 4.
An assumption complementary to multiplicative separability of the match value
function is that agents are randomly matched within each meeting place. Without
the assumption of random matching, the expected quality of a match in a meeting
place may be type dependent and may depend on the entire distribution of types in
the meeting place. In this case, the expected payoffs from joining a meeting place
would not be multiplicatively separable even if the match value function is, and
this would create the same kind of analytical difficulties as discussed earlier. For
example, if instead of one round of random matching we have sequential search as
in Burdett and Coles (1997) or in Damiano et al. (2005), the expected match quality for any type in a meeting place depends on which “class” the type belongs to.
Moreover, the class structure is endogenously determined by the type distributions
in the meeting place. How to incorporate sequential search into the framework of
price discrimination is an interesting and challenging topic that deserves further
research.
6 Appendix
Proof of Lemma 3.2 Suppose m (Tm ) is not a maximal pooling interval under qw .
There are two cases.
Case 1 Suppose that qw is not constant on m (Tm ). Then, we can find y, ỹ ∈
m (Tm ) such that qw (y) < qw (ỹ). It follows from condition ii) in Definition
2.1 that φw (y) ∩ φw (ỹ) = ∅ and m (φw (y)) ∩ m (φw (ỹ)) = ∅. Since Tm is
a maximal pooling interval and y, ỹ ∈ m (Tm ), we have E [t|t ∈ m (φw (y))] =
16
Endogenous reservation utilities will affect the monopolist’s exclusion policy. For example,
when the type distributions are symmetric and the common virtual type function J (t) crosses
zero only once, one can show that a free pool forces the matchmaker to increase market coverage.
This follows because to counter the competition by the free pool, the matchmaker needs to admit
more types at the bottom of the distribution so as to reduce the outside option for the participating
types.
E. Damiano and H. Li
E t|t ∈ m (φw (ỹ)) , which is possible only if inf m (φw (ỹ)) < sup m (φw (y)).
Then, there exist y1 ∈ m (φw (y)) and ỹ1 ∈ m (φw (ỹ)) such that y1 > ỹ1 . It follows that qw (y1 ) = qw (y) < qw (ỹ) = qw (ỹ1 ), which contradicts the assumption
that qw is nondecreasing.
Case 2 Suppose that there is a W ⊃ m (Tm ) such that qw is constant on W . By a
symmetric argument as in Case 1, we can show that qm is constant on w (W ). Since
W ⊃ m (Tm ), we can write w (W ) = w (m (Tm ))∪w (W \m (Tm )). We claim
that w (m (Tm )) ⊇ Tm : if x ∈ Tm , then there exists y ∈ m (Tm ) such that y ∈
φm (x), which by condition i) of Definition 2.1 implies that x ∈ φw (y), and therefore x ∈ w (m (Tm )). Further, w (W \ m (Tm )) = ∅ because W ⊃ m (Tm ),
and qw is constant and different from 0 on W . Finally, w (W \ m (Tm )) ∩ Tm = ∅,
because y ∈ m (Tm ) implies that φw (y) ∩ Tm = ∅ by condition i) of Definition
2.1. It follows that w (W ) ⊃ Tm . Since qm is constant over w (W ), we have
reached a contradiction to the assumption that Tm is a maximal pooling interval
under qm .
Proof of Lemma 3.3 We first establish that if x < inf Tm then sup φm (x) ≤ inf m
(Tm ). To prove this claim by contradiction, suppose that there exists y > inf m (Tm )
such that y ∈ φm (x). Since Tm is a maximal pooling interval and x ∈ Tm , we
have φm (x) ∩ m (Tm ) = ∅. By Lemma 3.2, m (Tm ) is an interval and hence
y ≥ sup m (Tm ). If inf φm (x) ≥ sup m (Tm ), then since qm (x) = E[y|y ∈ φm (x)]
and x < inf Tm , we have a contradiction to the assumption that qm is nondecreasing. If instead inf φm (x) ≤ inf m (Tm ), then there exists ỹ ∈ φm (x) such
that ỹ ≤ inf m (Tm ). By condition ii) of Definition 2.1 and the definition of
qw we have qw (y) = qw (ỹ). Monotonicity of qw implies that qw is constant on
[ỹ, y] ⊃ m (Tm ) therefore m (Tm ) is not a maximal pooling interval under qw ,
contradicting Lemma 3.2.
It follows from the above claim that φm (x) ⊆ m ([am , inf Tm )) for any x <
inf
T
G(inf m (Tm )) ≥
m , and hence [aw , inf m (Tm )] ⊇ m ([am , inf Tm )). Thus,
dG.
By
condition
iii)
of
Definition
2.1,
m ([am ,inf Tm )) dG =
m ([am ,inf Tm ))
dF,
which
implies
that
G(inf
(T
))
≥ F (inf Tm ).
m m
{x|φm (x)⊆m ([am ,inf Tm ))}
By a symmetric argument, we have sup φw (y) ≤ inf Tm for any y < inf m (Tm ).
Hence, [am , inf Tm ] ⊇ w ([aw , inf m (Tm ))) and F (inf Tm ) ≥ G(inf m (Tm )).
Then, we have G(inf m (Tm )) = F (inf Tm ), which by the definition of sm implies
that sm (inf Tm ) = inf m (Tm ). The argument for sm (sup Tm ) = sup m (Tm ) is
symmetric.
Proof of Lemma 3.4 Fix any sorting structure φ associated with q. We first show
that, if x does not belong to any maximal pooling interval, φm (x) is a singleton. Suppose y, ỹ ∈ φm (x) for some y < ỹ. By condition ii) of Definition 2.1,
φw (y) = φw (ỹ), and qw (y) = qw (ỹ). Since qw is monotone, it must be constant on
the interval [y, ỹ]. Therefore, there exists a maximal pooling interval W ⊇ [y, ỹ].
By construction, x belongs to w (W ) which is a maximal pooling interval by
Lemma 3.2; a contradiction.
Let φm (x) = {yx }. Since qm (x) = E[y|y ∈ φm (x)], we can write qm (x) = yx .
By monotonicity of qm , if x̃ < x and x̃ does not belong to a maximal pooling interval then yx̃ ≤ yx where φm (x̃) = {yx̃ }. Together with Lemma 3.3, this implies that
m [am , x] ⊆ [aw , φm (x)]. Clearly, yx does not belong to a maximal pooling interval under φw and φw (yx ) = {x}, so an identical argument yields w [aw , yx ] ⊆
Price discrimination and efficient matching
[am , x]. Then, by condition iii) in Definition 2.1, we have F (x) = G(yx ), or
qm (x) = yx = sm (x).
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